Theoretical Molecular Biophysics
"Theoretical Molecular Biophysics" is an advanced study book for students, shortly before or after completing undergraduate studies, in physics, chemistry or biology. It provides the tools for an understanding of elementary processes in biology, such as p
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In the early 1940s, Flory [6] and Huggins [7], both working independently, developed a theory based upon a simple lattice model that could be used to understand the nonideal nature of polymer solutions. We consider a lattice model where the lattice sites are chosen to be the size of a solvent molecule and where all lattice sites are occupied by one molecule [1].
2.1 Monomeric Solution As the simplest example, consider the mixing of a low-molecular-weight solvent (component α) with a low-molecular-weight solute (component β). The solute molecule is assumed to be the same size as a solvent molecule and therefore every lattice site is occupied by one solvent molecule or by one solute molecule at a given time (Fig. 2.1). The increase in entropy ΔSm due to mixing of solvent and solute is given by N! , (2.1) ΔSm = kB ln Ω = kB ln Nα !Nβ ! where N = Nα + Nβ is the total number of lattice sites. Using Stirling’s approximation leads to ΔSm = kB (N ln N − N − Nα ln Nα + Nα − Nβ ln Nβ + Nβ ) = kB (N ln N − Nα ln Nα − Nβ ln Nβ ) Nα Nβ − kB Nβ ln . = −kB Nα ln N N
(2.2)
Inserting the volume fractions φα =
Nα , Nα + Nβ
φβ =
Nβ , Nα + Nβ
the mixing entropy can be written in the well-known form
(2.3)
20
2 Flory–Huggins Theory for Biopolymer Solutions
solvent solute
Fig. 2.1. Two-dimensional Flory–Huggins lattice
ΔSm = −N kB (φα ln φα + φβ ln φβ ).
(2.4)
Neglecting boundary effects (or using periodic b.c.), the number of nearest neighbor pairs is (c is the coordination number) c Nnn = N . 2
(2.5)
These are divided into Nα c N φ2α c φα = , 2 2 = N φα φβ c.
Nαα = Nαβ
Nββ =
N φ2β c Nβ c φβ = , 2 2 (2.6)
The average interaction energy is w=
1 1 N cφ2α wαα + N cφ2β wββ + N cφα φβ wαβ 2 2
(2.7)
which after the substitution wαβ =
1 (wαα + wββ − w) 2
(2.8)
becomes 1 w = − N cφα φβ w 2 1 1 = + N cφα (φα wαα + φβ wαα ) + N cφβ (φβ wββ + φα wββ ) 2 2
(2.9)
and since φα + φβ = 1 1 1 w = − N cφα φβ w + N c(φα wαα + φβ wββ ). 2 2
(2.10)
Now the partition function is N! Nα !Nβ ! N! . = (zα e−cwαα /2kB T )Nα (zβ e−cwββ /2kB T )Nβ eNα Nβ cw/2N kB T Nα !Nβ ! (2.11)
Z = zαNα zβ β e−Nα cwαα /2kB T e−Nβ cwββ /2kB T eNα Nβ cw/2N kB T N
ΔFm /NkBT
2.1 Monomeric Solution 0
χ =2
–0.2
χ =2
–0.4
χ=1
–0.6
χ =0 0
0.2
0.4
0.6
0.8
21
1
φα
Fig. 2.2. Free energy change ΔF/N kB T of a binary mixture with interaction
The free energy is F = −kB T ln Z = −Nα kB T ln zα − Nβ kB T ln zβ Nα Nβ cw c c + N kB T (φα ln φα + φβ ln φβ ). = +Nα wαα + Nβ wββ − 2 2 2N (2.12) For the pure solvent, the free energy is c F (Nα = N, Nβ = 0) = −Nα kB T ln zα + Nα wαα 2
(2.13)
and for the pure solute c F (Nα = 0, Nβ = N ) = −Nβ kB T ln zβ + Nβ wββ . 2
(2.14)
Hence the change in free energy is ΔFm = −
Nα Nβ cw + N kB T (φα ln φα + φβ ln φβ ) 2N
(2.15)
with the energy change (van Laar heat of mixing) ΔEm = −
cw Nα Nβ cw = −N φα φβ = N kB T χφα φβ . 2N 2
(2.16)
The last equation defines the Flory interaction parameter (Fig. 2.2): χ=−
cw . 2kB T
(2.17)
For χ > 2, the free energy has two minima and two stable phases exist. This i
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